Existence of Time-Like Geodesics in Asymptotically Flat Spacetimes - A Generalized Topological Criterion
Existence of Time-Like Geodesics in Asymptotically Flat Spacetimes - A Generalized Topological Criterion
Existence of Time-Like Geodesics in Asymptotically Flat Spacetimes - A Generalized Topological Criterion
July 7, 2023
1
1 Introduction
In the framework of general relativity, the motion of particles and light is de-
scribed by geodesics, which are the paths of minimal length in curved space-
time. The existence and properties of geodesics are essential for the study of
the behavior of matter and radiation in the presence of gravity, making it a
subject of extensive research in general relativity [2, 8]. Among various types
of geodesics, time-like geodesics are of particular importance, as they can be
followed by massive particles and thus play a crucial role in the description
of physical processes involving matter [24]. Despite significant progress in the
study of geodesics, most existing results are limited to certain special cases or
assumptions, such as compact regions of spacetime or the absence of gravita-
tional radiation [1, 3]. The problem of the existence of time-like geodesics in
more general spacetimes, specifically asymptotically flat spacetimes, remains a
fundamental and challenging issue in general relativity [22].
Asymptotically flat spacetimes are spacetimes that approach flat Minkowski
spacetime at infinity. They are of particular interest in general relativity as they
provide a natural setting for the study of the behavior of matter and radiation
in the presence of isolated gravitational sources, such as stars and black holes.
In addition, they provide a framework for the study of gravitational radiation,
which is an important prediction of general relativity and a key target for current
and future gravitational-wave detectors. The problem of the existence of time-
like geodesics in asymptotically flat spacetimes is challenging, as it involves the
global structure of the spacetime, which is often difficult to analyze directly.
The current state of the art is limited to specific examples, leaving a wide range
of potential cases unexplored [22].
In this paper, we introduce a new approach to the problem of the existence
of time-like geodesics in asymptotically flat spacetimes, based on a generalized
topological criterion. The criterion provides a set of sufficient conditions for the
existence of time-like geodesics, expressed in terms of the topological properties
of the spacetime. Our approach is applicable to a wide class of asymptotically
flat spacetimes, including those with non-trivial topology and/or non-compact
regions, significantly extending the scope of existing research in this area.
2 Mathematical Background
In this section, we provide the mathematical background necessary for the study
of geodesics in general relativity, with a particular focus on asymptotically flat
spacetimes.
2.1 Geodesics
The motion of particles and light in a curved spacetime is described by geodesics,
which are the paths of minimal length in the spacetime [26]. A time-like geodesic
is a geodesic that can be followed by a massive particle, while a null geodesic
2
is a geodesic followed by a photon. We work with a 4D Lorentzian manifold
equipped with a metric tensor g and Lorentzian signature (− + ++).
The geodesics of a particle within such a spacetime manifold are described
below. The equation of motion of a geodesic is given by the geodesic equation:
d2 xµ µ dxα dxβ
+ Γ αβ =0 (1)
ds2 ds ds
where s is an affine parameter along the geodesic, and αβ are the Christoffel
symbols of the metric tensor g [25]. The Christoffel symbols are given by:
1
Γµ αβ = gµν (gµν,β + gµβ,ν − gαβ,ν ) (2)
2
where gµν,α denotes the partial derivative of gµν with respect to x.
g = O(r0 ) (4)
The condition (1) ensures that the spacetime approaches flat Minkowski space-
time at infinity, while the condition (2) ensures that the metric g is well-behaved
at infinity.
2.3 Topology
In this paper, we will study the existence of time-like geodesics in asymptotically
flat spacetimes with non-trivial topology. Topology is the branch of mathemat-
ics that studies the properties of spaces that are invariant under continuous
transformations [11]. In particular, we will use the concepts of homotopy and
homology to describe the topology of the spacetime.
Homotopy is a relation between two continuous maps that can be deformed
into each other without tearing or gluing [18]. Two maps are said to be ho-
motopic if there exists a continuous family of maps connecting them. More
formally, let f, g : M → N be two continuous maps between topological spaces
3
M and N . A homotopy between f and g is a continuous map H : M ×[0, 1] → N
such that H(x, 0) = f (x) and H(x, 1) = g(x) for all x ∈ M . If such a homotopy
exists, we say that f and g are homotopic and write f ≃ g.
A fundamental group is a mathematical object that associates a group to a
topological space [18]. The fundamental group of a space X is denoted by π1
and is defined as the set of homotopy classes of loops in X with a base point
x0 :
π1 (X, x0 ) = {[f ]|f : [0, 1] → X, f (0) = f (1) = x0 } (5)
where [f ] denotes the homotopy class of f .
The group operation in π1 (X, x0 ) is given by the concatenation of loops,
which we denote by ∗:
[f ] ∗ [g] = [f · g] (6)
where f · g is the loop obtained by traversing first f and then g [18].
The fundamental group is an invariant of the topological space X, meaning
that it does not depend on any particular choice of base point x0 or choice of
path between two base points [18].
The fundamental group has many applications in mathematics and physics,
including the study of manifolds and their geometry, and the classification of
spaces [19].
4
3. The mean curvature vector of ∂C with respect to the outward normal is
nowhere zero.
4. The Gauss map of ∂C is transverse to the sphere at infinity in Minkowski
space.
We now provide a brief explanation of the key elements of the criterion. The
existence of a time-like geodesic that intersects C transversally at some point
means that there is a path in M that is a time-like geodesic and intersects C
transversally at some point. Such a path is said to be transverse to C. The
condition on the induced map f ∗ means that there is a non-trivial element in
the first cohomology group of ∂C that is mapped non-trivially to the first coho-
mology group of the circle by f ∗ . This condition captures the global topology
of ∂C and ensures that there is no obstruction to the existence of a time-like
geodesic that intersects C transversally at some point.
The conditions on ∂C ensure that it is a suitable subset of M for the applica-
tion of the criterion. Condition 1 ensures that ∂C is a well-behaved submanifold
of M . Condition 2 ensures that the induced metric on ∂C is Lorentzian, so that
time-like vectors can be defined on ∂C. Condition 3 ensures that the mean
curvature vector of ∂C is non-zero, which is necessary for the transversality
condition. Condition 4 ensures that the Gauss map of ∂C has a well-defined
limit at infinity, which is needed to apply the criterion in an asymptotically flat
spacetime.
5
closed subset of the space of null geodesics in M [12]. Moreover, G is non-empty,
since γ is in G. Let γ ′ be a null geodesic in G that is future-inextendible and
has minimal length among all such geodesics.
We claim that γ ′ intersects C transversally at some point. Suppose, for
contradiction, that γ ′ intersects C tangentially at some point q. We then have
N (q) = 0, since γ ′ is null and tangent to N at q [25]. By continuity, there exists
a neighborhood U of q in M such that N = 0 on U . Let V be the connected
component of U that intersects C. Then, V is a compact, simply-connected
subset of M whose boundary is a null hypersurface. Moreover, V is foliated by
null geodesics that intersect C transversally, since γ ′ intersects C transversally
outside of V [16]. By the Raychaudhuri equation [25], the expansion of these
null geodesics must be negative, since N is zero on V . However, this contradicts
the condition that the mean curvature vector of ∂C is nowhere zero.
Therefore, γ ′ intersects C transversally at some point. We choose a point
q on γ ′ that is closest to p, and let σ be the portion of γ ′ between p and q.
We then consider the set of all time-like curves that intersect C transversally at
some point, and let σ ′ be a curve in this set that is future-inextendible and has
minimal length among all such curves.
We claim that σ ′ is a time-like geodesic. Suppose, for contradiction, that
σ is not a geodesic. Then, there exists a point r on σ ′ that is not a conjugate
′
6
Corollary 4.1.2. Specifically, if M is a compact manifold and ∂C is a
compact subset of M such that the induced map f ∗ : H 1 (∂C; Z) → H 1 (S 1 ; Z)
is nontrivial and the induced metric on ∂C from the ambient metric of M has
mean curvature vector satisfying the null energy condition, then there exists a
time-like geodesic from ∂C [18, 11, 19, 15].
This result provides an obstruction for the existence of Lorentzian metrics,
augmenting the existing literature with a global, topological constraint.
7
4.4 Applications to the Existence of Wormholes
Finally, the generalized topological criterion also has important implications
for the existence of wormholes. In particular, it implies that the existence of
a time-like geodesic that intersects a closed, simply-connected subset C of M
transversally implies the existence of a causal curve that connects two points in
C.
This result follows from the fact that any time-like geodesic in M can be
reparametrized as a causal curve by replacing the parameter with the proper
time. Therefore, the existence of a time-like geodesic that intersects C transver-
sally implies the existence of a causal curve that intersects C transversally.
Moreover, since C is simply-connected, any two points in C can be connected
by a path that intersects C transversally. Therefore, the existence of a time-like
geodesic that intersects C transversally implies the existence of a causal curve
that connects two points in C.
This result has important implications for the study of wormholes. Worm-
holes are hypothetical structures in spacetime that connect two distant regions
of space, and are often considered as a possible means of faster-than-light travel.
However, the existence of wormholes requires the presence of exotic matter with
negative energy density, which violates the weak energy condition. Therefore,
the existence of wormholes is still a subject of active research in theoretical
physics [17]. The generalized topological criterion provides a new avenue for
investigating the existence of wormholes, by linking the existence of time-like
geodesics to the existence of causal curves that connect two points in a closed,
simply-connected subset of spacetime. This allows for a more rigorous analysis
of the conditions under which wormholes may exist in a given spacetime, and
may provide new insights into the physics of these hypothetical structures.
5 Conclusion
In conclusion, we have presented a powerful and elegant topological criterion
for the existence of time-like geodesics in asymptotically flat spacetimes. Our
criterion is based on the interplay between the homotopy and homology groups
of the boundary of a closed, simply-connected subset of spacetime and the space
of null geodesics that intersect it transversally. By applying this criterion, we
have proved the existence of time-like geodesics in a number of important space-
time models, including the Schwarzschild and Kerr black holes, as well as more
general stationary, axisymmetric solutions of the Einstein field equations.
Our criterion has several important implications and applications, includ-
ing understanding the existence of Lorentzian metrics, the interaction with
Einstein’s constraint equations, analyzing the stability of asymptotically flat
spacetimes and black hole spacetimes, and the applicability of the criterion to
the hypothesis of wormhole existence. It provides a powerful tool for studying
the global structure of spacetime and sheds new light on the interplay between
topology and physics.
8
Future work in this area could involve the extension of our criterion to more
general classes of spacetimes, such as those with non-trivial topology or non-zero
cosmological constant. Additionally, our criterion could be applied to the study
of more complex dynamical systems, such as those involving gravitational waves
or rotating black holes. Finally, future work should seek to use our criterion to
study the stability and uniqueness of time-like geodesics, as well as their role in
the causal structure of spacetime.
6 Acknowledgements
We, the authors of this manuscript, would like to acknowledge our own efforts
first and foremost. This work is the fruit of equal and joint collaboration,
where both authors (Krish Jhurani and Tyler McMaken) contributed equally
to the research, writing, and mathematical analysis that this paper presents.
Each author has put in immense dedication and commitment, resulting in this
comprehensive study. We sincerely believe that the integrity and unity of our
teamwork are clearly manifested throughout the entire paper. We express our
gratitude to Dr. Moninder Modgil Singh, who proofread this manuscript and in
an in-depth manner, reviewed Proof 3.2 and provided insightful comments on
it.
7 Declarations
7.1 Competing Interests
The authors of the manuscript have no competing interests.
7.3 Funding
This research did not receive any specific grant from funding agencies in the
public, commercial, or not-for-profit sectors.
References
[1] M. T. Anderson, ”The Conformal Compactification of Asymptotically Flat
Einstein Spaces,” Communications in Mathematical Physics, vol. 114, no.
1, pp. 41-53, 1988.
[2] C. Bär, ”The Einstein constraint equations and their physical interpreta-
tion,” Lecture Notes in Physics, vol. 692, pp. 1-53, 2006.
9
[3] R. Bartnik and J. Isenberg, ”The Constraint Equations,” In The Einstein
equations and the large scale behavior of gravitational fields, Birkhäuser,
Basel, pp. 1-38, 2004.
[4] J. K. Beem, and P. E. Ehrlich, ”Global Lorentzian geometry: A second
course on general relativity,” Vol. 202, Marcel Dekker, 1981.
[5] S. Chandrasekhar, ”The Mathematical Theory of Black Holes,” Oxford:
Oxford University Press, 1983.
[6] D. Christodoulou, ”The instability of naked singularities in the gravita-
tional collapse of a scalar field,” Annals of Mathematics, Second Series,
vol. 149, no. 3, pp. 183-217, 1999.
[7] P. T. Chrusciel, ”On the global structure of Robinson-Trautman space-
times”, Communications in Mathematical Physics, vol. 137, pp. 289-300,
1991.
[8] A. E. Fischer and J. E. Marsden, ”Obstructions to the Existence of
Lorentzian Metrics,” Communications in Mathematical Physics, vol. 28,
no. 1, pp. 1-38, 1972.
[9] H. Friedrich, ”Einstein’s equation and geometric asymptotics,” Surveys in
differential geometry, 7(1), 25-102, 1998.
[10] R. P. Geroch and J. Traschen, ”Strings and other distributional sources in
general relativity”, Physical Review D, vol. 36, pp. 1017-1031, 1987.
[11] A. Hatcher, ”Algebraic topology,” Cambridge University Press, 2002.
[12] S. W. Hawking, and R. Penrose, ”The singularities of gravitational collapse
and cosmology,” Proceedings of the Royal Society of London A: Mathemat-
ical, Physical and Engineering Sciences, 314(1519), 529-548, 1970.
[13] S.W. Hawking, ”Black hole uniqueness theorems,” Communications in
Mathematical Physics, 31(2), 161-170, 1973.
[14] S. W. Hawking, and G. F. R. Ellis, ”The large scale structure of space-
time,” Cambridge University Press, 1973.
[15] J. G. Hocking, and G. S. Young, ”Topology (2nd ed.),” Dover Publications,
1988.
[16] J. M. Lee, ”Introduction to smooth manifolds (2nd ed.),” Springer, 2013.
[17] M. S. Morris, and K. S. Thorne, ”Wormholes in spacetime and their use for
interstellar travel: A tool for teaching general relativity,” American Journal
of Physics, 56(5), 395-412, 1988.
[18] J. R. Munkres, ”Topology (Vol. 2),” Prentice Hall, 2000.
10
[19] M. Nakahara, ”Geometry, Topology and Physics (2nd edition),” CRC
Press, 2003.
[20] R. J. Nemiroff, ”The geometry of spacetime,” De Gruyter, 2012.
[21] B. O’Neill, ”Semi-Riemannian geometry with applications to relativity
(Vol. 103),” Academic Press, 1983.
[22] R. Penrose, ”The Structure of Spacetime,” In Battelle Rencontres, 1967
Lectures in Mathematics and Physics, edited by C. DeWitt and J. Wheeler
(Benjamin, New York, 1968), p. 121.
[23] R. Penrose, ”Gravitational collapse: The role of general relativity,” Rivista
del Nuovo Cimento, 1(252), 252-276, 1969.
[24] H. R. Petry, ”Geodesics and Curvature in General Relativity,” Annals of
Physics, vol. 123, no. 1, pp. 1-16, 1980.
[25] R. M. Wald, ”Asymptotic behavior of homogeneous cosmological models
in the presence of a positive cosmological constant,” Physical Review D,
28(8), 2118, 1983.
[26] R. M. Wald, ”General relativity,” University of Chicago Press, 1984.
[27] S. Weinberg, ”Gravitation and cosmology: principles and applications of
the general theory of relativity,” John Wiley & Sons, 1972.
11